Reference for proof of an integral from the "Tables of Integral Transforms" involving a Gaussian and a Laguerre polynomial

Question

I am looking for a proof of one of the integrals presented in Harry Bateman's Tables of Integral Transforms. The specific integral in question is presented on page 42 in chapter 8.9 as equation (3):

$$ \int_0^{\infty} \left[x^{\nu+\frac{1}{2}} \mathrm{e}^{-\frac{1}{2} x^2} L^{\nu}_n\!\left(x^2\right) \right]J_{\nu}\!\left(xy\right) \sqrt{xy}\,\mathrm{d}x = \left(-1\right)^n \mathrm{e}^{-\frac{1}{2} y^2} y^{\nu+\frac{1}{2}} L^{\nu}_n\!\left(y^2\right) $$

where $J_{\nu}$ are the Bessel functions of the first kind and $L^{\nu}_n$ are the associated Laguerre polynomials. This integral is basically the Hankel transform of the function $f\!\left(x\right) = x^{\nu} \mathrm{e}^{-\frac{1}{2} x^2} L^{\nu}_n\!\left(x^2\right) $ with an additional factor $\sqrt{y}$ (depending on the definition).

A similar integral also appears in this question, but the integral in question here is a bit different.

The full reference of the table entry is:
Harry Bateman, Tables of Integral Transforms, Volume 2, p. 42 (ch. 8.9, eq. (3))

Answer 1

The question asks for a reference; the body asks for proof. Here's a proof that's really more of a verification. From the proposed formula make a generating formula: $$\sum_{n=0}^\infty z^n \int_0^\infty x^{\nu+1}\exp{(-x^2/2)}L_n^{\nu}(x^2) J_{\nu}(xy)\,dx \overset{?}{=}$$ $$ y^{\nu}\exp{(-y^2/2)} \sum_{n=0}^\infty (-z)^n L_n^{\nu}(x^2).$$ Use the generating formula for Laguerre polynomials (Gradshteyn and Rhyzik 8.975.1), $$\sum_{n=0}^\infty z^n L_n^{\nu}(x) = (1-z)^{-\nu-1}\exp[-x \ z/(1-z) ].$$ Use G&R 6.631.4 $$\int_0^\infty x^{\nu+1}\exp{(-ax^2)}J_{\nu}(xy)dx = \left( \frac{y}{2a} \right)^{\nu} \frac{1}{2a} \exp\left[-y^2/(2a)\right].$$ The prior two formulas are well known. Algebra completes the verification. For a proof, work backwards.